Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

What does “College Readiness” really mean for math class? I mean… really…

This post has questions. No answers in this post. Just questions.

I just spent the better part of a day exploring the SAT test (which Michigan has recently adopted as the test that all high school juniors will take as a “college readiness” test.)

I also read this Slate article which speaks to people who feel like they aren’t “math people.”

As I continue to listen to various groups chime in with that they think kids need with respect to the mathematics portions of our various educational systems, there seem to be a few ideas that are coming out.

The first two are usually close-to-unanimously agreed upon.

  1. All young people… ALL… young people have a God-given right to a high-quality education.
  2. A high-quality education includes a significant amount of mathematics beyond basic numeracy.

Beyond that, the overarching ideas push into value-based philosophies about what the author or speaker believes are in the best interest of American young people. These are definitely thoughts where two reasonable people could find areas of disagreement.

3.  An education earns the title “high-quality” when the receiver can use it to                    successfully take the desired next steps after it’s done. “Next steps” are                      generally considered one of the following: A. going to college, B. going to                    work, C. going into the military, D. starting a family or E. any combination                    thereof. No one necessarily more noble or challenging than the others.

4. The overseers of those areas are the authority on what is required to be                     able to successfully join those communities. College professors, business                   leaders, military leaders, and church leaders all have a reasonable                             expectation that they would influence the courses of study that lead into                     those individual arenas.

If we push these thoughts to the next level, it becomes reasonable to assume that college professors, business leaders, military leaders and church leaders are going to quite often disagree on the necessary requirements for an education to be considered “successful”. What’s more, there will be large segments of the general public who would prefer one or two of those get a larger say than the others. Certain groups of people would prefer that college professors should have the last say. The incredible number of religious private schools speak volumes to our public’s desire to allow their particular faith to have the ultimate say in the educational program.

And math doesn’t get a free pass in these disagreements. You will see wide differences in the types of mathematical content that are preferred as well as the instructional and assessment methods. The nationwide introduction of Common Core has demonstrated that math curriculum can trigger some very negative responses from significant portions of the American public.

And the fact that nationwide, our school systems are “public” empowers the general public to have a say. In fact, this is often a tricky balancing act for education professionals. There is a certain amount of background knowledge necessary to make sound decisions within the field of education. But, practically every single adult walking the streets has a decade (give-or-take a year or two) of experience within the field of education. There is a certain amount of boldness that that kind of familiarity breeds.

Result: Everyone has an opinion on how education should look. And that opinion is largely based on the relative satisfaction that the opinion-bearer feels when he/she reflects on his/her past experiences.

That is a very good thing with some inconvenient consequences. One of those inconvenient consequences arises when legislators get involved. The American public has an oddly-trusting, yet often cynical relationship with their elected officials. Most people have very few good things to say about them. But when it comes to their own personal beliefs about society, getting their views enshrined into law become the highest priority. You can see this on both sides of the political spectrum. These institutions disgust us and we don’t trust them, but we want them on our side. It’s an odd paradox.

This paradox extends into the fields of education where, at least here in Michigan, the state government has made several plays that are tipping the scales in terms of which of the aforementioned groups is getting the state favor. As a result, “College-and-Career Readiness” is becoming cliche.

But our familiarity with it doesn’t mean that we have any idea what to do with it. Moreover, it doesn’t even mean that the general public has agreed upon definitions of “college ready” or “career ready.” The state has just solidified those two arenas as the goals.

It’s our job as educators to figure out how we are setting up our classrooms, schools, and districts to maximize our impact on young people toward those goals.

In my next post, I am going to lay out the primary issue that is creating this inner conflict I feel…


We’re not just teaching math…

The old adage goes “I don’t teach math. I teach children.” That line typically gets used when an educator’s focus is a bit out of balance with respect to empathetic student-centered attitudes and content-driven, fidelity-to-curriculum attitudes. There needs to be a balance and it can be tricky to find sometimes.

In addition to that, there is another balance that needs to be struck. The balance between the math content in a curriculum and the other skills the students are going to need to learn the math content. Some of these skills are considered “soft skills” by some. These are things like communication skills, presentations, research, teamwork. I’ve always been a bit uncomfortable with the term “soft skills”. (We can talk more about that another time if you want).

Beyond those, there are some “hard” skills that some math teachers just feel isn’t their job to teach. These are things like technology skills, reading, writing, and supplementary (often much lower-level) math skills. I’ve been in a variety of math classrooms talking to teachers of high school math who feel like they just shouldn’t have to teach fractions, long division, and reading.

Yet, increasingly math classes are starting to look like this.

2014-02-20 10.41.53

We’re not just teaching math…

And those skills, be them “soft” or “hard” will directly impact our students’ ability to learn the math content that we are hoping they’ll learn. I think it is important that we math teachers simply expect to have to teach our students to do everything we need them to do to be successful in our math classes.

And this includes remembering that teaching and learning have some recurring patterns when done successfully: teacher modeling, student exploration, student individual practice, formative assessment, feedback. These are things that exist in every successful math class I’ve seen. (Depending on teacher philosophy, the order of the steps might not be the same in every classroom, but the steps are all still there.)

Very few teachers will tell you that you can skip that teaching-and-learning process for the math content.

Many more will skip that teaching-and-learning process with the “softer” parts of their curriculum.

Perhaps, I should back up and discuss how I see “curriculum”. From the teacher perspective, curriculum includes both the “what” of the learning, but also the “how”. And if a math teacher has students whose math experience looks like this…

2013-10-24 13.04.22

… as is increasingly becoming the case, then the curriculum probably includes three fairly broad categories.

Math Content: This would include the primary learning targets for the course, but also the prerequisite math knowledge that the students need to advance successfully to the new content.

Learning Tools: Depending on the class this might include a couple of devices (calculators, iOS devices, laptops, Chromebooks) and any other manipulatives (Alge-blocks, patty paper, compasses, protractors, etc.). If the learning will require the use of these tools, then the learning of these tools is every bit as much a learning objective as the math content.

Classroom Procedures: Where will the schedule be posted? Where will handouts be made available? How does a student turn in assignments? Where should a student look when he/she has been absent? What does a student do when they are trying to work at home and find themselves paralyzed by confusion?

If a student struggles to learn well the content in any of those three areas, that student will start riding the struggle bus pretty quickly. The first step to avoid this is to recognize that we are going to have to actively teach all of the things students need to know to be successful in our class. How many of us run a formative assessment where the learning target of the assessment is “Students will be able to use a compass”? How many of us give feedback on the learning target “students will know how to create table with ordered pairs on Desmos”?

Remember, we don’t teach math. We teach children to learn math. And that requires us math teachers to remember that there’s actually a lot more than math knowledge that students will need to be successful in our classes.

The effective leader has all the right… questions.

effective leader

There’s plenty that’s been said about effective school leadership. (Here, I’ll save you a couple of key strokes.) I don’t think that I am going to share anything revolutionary here.

But I want to share an anecdote that I heard recently from a trusted colleague that I thought spoke very clearly to the power of an effective leader. As he reflected aloud on his first couple of years teaching, he made a couple of statements that I found to be very powerful. (I’m using quotes, but this is certainly paraphrased.)

“I was blessed to have a principal that asked a lot of questions. It wasn’t that I was doing wrong things or bad things, but it showed me that I was doing a lot of things without having a very good reason.”

It was an outstanding summary of the role of mentoring a new teacher. Teacher prep programs in most universities fall short of their goals. It isn’t necessarily their fault. The teaching profession largely sets new teachers up to be steamrolled and as the model currently exists (a ton of content courses, a few professional prep courses, a short internship and go forth and prosper), it would be pert-near impossible for any new teacher to enter fully prepared.

So, most teachers fall back on what “teachers do.” But why? And if you don’t know why, then (at the very least) leaders need to make sure they can get that far. It might be that talking at the board is the right move for that young math teacher. It could also being fully blended with instructional technology would be better. But the teacher needs to know why. What goals will that meet? What content will that work best for? How are you making sure it’s effective in meeting its goals?

We should all be blessed to have leaders that ask a lot of questions. I don’t think it’s enough to have everyone doing the right things. It’s good. But people who do the right things without knowing why can’t reflect on their effectiveness, they can’t be flexible within the systems, they can’t roll with the punches when the results don’t appear as perfectly as they should.

There are a lot of excellent teachers making a lot of excellent innovations and building a lot of excellent systems in their classrooms and in their school communities. What makes them excellent is that they know what problem they are trying to address and they have the means to verify whether or not the innovation is solving the problem. It’s effective because they know the why.

And rather than instructional skills or techniques, perhaps it’s more important that leaders lead their people to be able to think about their classrooms like that.

And reminding myself that teachers are the leader of their classrooms, if instilling this thought process is so powerful for young, developing teachers, what would it look like for teachers to instill this in young, developing students?

Teachers: At what are you an expert?

This is the second in a series of reflections that came out of a fantastic sit-down with #MichED -ucators Melody Arabo (@melodyarabo) and Jeremy Tuller (@jertuller). Melody asked a question that followed up by mentioning that teachers have a really, really hard time answering: What parts of your professional work would you consider yourself to be an expert?

You see, the teaching profession makes it’s members uneasy by self-promotion. And it’s understandable. Teaching is a complex skill set. Teachers are renowned for having very, very broad sets of abilities as posters like this indicate:

Just a teacher

Technology adds even more lines to this poster. So, with so many different nooks and angles to the work, it can be very understandable that teaching is a profession that makes it’s practitioners feel as though their efforts are stretched a mile wide and an inch thick. It’s hard to feel like an expert at anything under those circumstances.

But we need to. We need our expert teachers to not only be aware of their areas of expertise, but also be willing to advertise it. There’s a lot of teachers in this state. Lots. Like… tens of thousands. It shouldn’t scandalize us that each teacher has strengths and weaknesses. And some teachers have lots and lots of strengths. Every profession has it’s hall-of-famers. I could name a few that I’d nominate for a public school teaching hall-of-fame. Duane Seastrom… Eileen Slider… (Did you just think of a couple that you’d nominate?)

And teachers have a darn good perspective on this. They know who the good ones are and what they are so good at. “Kids never act out for her.” “The projects they do for him are amazing.” “She gets amazing growth out of students with disabilities.” But chances are, those teachers aren’t blogging about it. Chances are they don’t have business cards that say, “Mrs. Taylor, instructional designer, classroom manager.” Chances are they aren’t promoting the practices that they use that work. Chances are they aren’t standing up in staff meetings showing 5-min video clips of the awesome things their students are doing. Because teachers don’t do that.

What is it about teaching that makes it’s practitioners uncomfortable declaring their strengths and advertising them?

I have my own half-baked ideas. (Comment opportunities for dissent, if you’re in the mood.) For one, teacher evaluations are really time-consuming and we really haven’t figured out how to do it yet. What are the best practices? How important is student achievement? How do you measure positive impact of a teacher on an unsuccessful student? These are really, really tough things to measure. This is a symptom of our inability to collectively agree on the exact role that the teacher plays in the education industry. I can all think of the teacher whose in-class practice is pretty good, but it completely uninvolved in the community. I can also think of teachers whose instructional and assessment practices aren’t stellar, but they do a wonderful, wonderful job of reaching out to the marginalized students and keep them coming to school. I can also think of teachers who are inept in supporting the struggling students in their classroom, but because they coach three sports help keep an different population of struggling students eligible so that they can stay active on their teams. All three of those teachers are playing roles that are tough to evaluate. Obviously, we want every teacher to be hall-of-fame quality at instruction and assessment, but how do you isolate the “mandatory” skill set?

From the other side, collective bargaining has put pressure on teachers to not really separate themselves in any major way. If we find ourselves with a handful of teachers who are exemplary teachers, it’s very short logical leap to those teachers deserving some sort of reward for being so good at what they do. Naturally, there’s a very reasonable desire on a whole lot of different levels to shield the teaching profession from this. We want this field to become more collaborative. Not more competitive. So, to protect against that the unions have stayed far, far away from emphasizing outspoken greatness of individual teachers.

That doesn’t mean that greatness doesn’t exist. It just means that greatness is staying contained. We don’t want that. We want greatness to spread. And given the technology, given the pressure, given the difficulty of being a teacher, it is becoming more and more reasonable for teachers to begin identifying the exemplary practitioners and trying to figure out what of their skills can be displayed and transferred.

There are three things that I believe to be true. 1. It’s possible for every teacher to be a great teacher. 2. Not every teacher is currently a great teacher. And 3. It is in the best interest of every student in America to be in the classroom of a great teacher just as often as possible.

So, I’ll ask again: What parts of your professional practice would you consider yourself to be an expert at? What do you do really, really well that you could demonstrate to mentor up a young or struggling teacher? What are the things you do that are so good that you’d be willing to share them on the open educational marketplace of ideas? Feel free to reply in the comments section. That way if you have a weakness in the area that matches with a person’s stated strength, you can reach out to them and open that conversation.